Answered: Problem #4: Let u = (72, 1, 0). Find… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 31E

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Problem #4: Let u = (72, 1, 0). Find the orthogonal projection of u onto the subspace W spanned by the (non-orthogonal)
vectors u₁ = (1,-1,0) and u₂ = (0, 12, -1).
Problem #4:
(A) (15¹, 1,-6) (B) (¹431-6) (C) (¹47, 1-6) (D) (¹451-6) (E) (14⁹,-6) (F) (157, 11, −6)
(G) (153, 1-6) (H) (1556)
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Find a basis for R2 that includes the vector (2,2).
Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.
3. (a) Let A = (b) Let A = 1 1 0 1 1 1 0 1 1 1 1 combination of the column vectors of A, with coefficeints x1, x2, and x3. (c) Now generalize part (b) to the case where A is m × n. That is, show that if A is an 1 1 0 1 1 1 1 0 1 1 1 18 - What are the row vectors and column vectors of this matrix? 9 arbitrary mx n matrix and x = m and let = X1 x2 x 3 X1 x2 In the columns of A, with coefficients x1, x2,., n. be a vector in R³. Show that A is a linear 2c1 c₂ + c3, and let (d) Suppose that A is a 4 × 3 matrix, and let c₁, c2, c3 be the column vectors of A. That A [C₁ C₂ C3]. Suppose that be R4 is the vector 6 = 20₁ [ci is, is in R", then A is a linear combination of 9 X1 = X2 . Show that Aỡ = b is consistent by finding a solution to the system.
1. (a) Find the matrix P that multiplies every vector (x, y, z) E R³ to produce the vector (3x+2y+z, 5y- z, 8x). Also find P-1. (b) Find the matrix P that multiplies every vector (x, y) E R2 to produce (5x-4y, - 2x, 3y - 2x) = R³
Problem #4: Let u = (18,1,0). Find the orthogonal projection of u onto the subspace W spanned by the (non-orthogonal) vectors u₁ = (1,-1,0) and u₂ E (0,6,-1). (A) (4-3) (B) (1.-3) (C) (3) D) (27.1.-3) )(2-3) () (3/5, 1/, -3) (G) (2.-3) (27.1, -3)
I need to solve this problem without using matrices. We have a vector V whose components in the basis B = (v1,v2) are (3,-2) Which are the components of the same vector in the basis C = (u1,u2) if v1 = 4u1-u2 and v2 = 4u1-3u2? Remember, no matrices.
(a) Draw the vectors a = (4, 3), b = (2, -1), and c = (9, 1). -10 -10 10r 10 10 k p ku p -5 10 -5 5 10 -10 5 10 -10 -5 b 5 10 -5 -5 -5 -105 y -10 (c) Find the exact values of s and t. -10 -5 10 (b) There are scalars s and t such that c = sa + tb. Use a sketch to estimate the values of s and t. (Round your answers to one decimal place.) S% -10 y
Problem 5 Consider the set of vectors 1 S = 1 3 -2 (i) Determine if this set of vectors is linearly independent or linearly dependent. (ii) If the set is linearly dependent, determine a linear dependent relationship between these vectors. (iii) Consider the transformation xi + 2x2 + x3 T x2 2^1 + За2 + 2з + 2 X3 3x1 + x2 – 2x3 Determine if T is a linear transformation or not.
9. Find all real values of c such that the vectors and are linearly independent.
2. The vector x = H [13] vector [2]B = [8] A 1 B 2 3 D0 in R2 relative to the orthogonal basis B What is the value of a + b? {0-[-]} has coordinate
(5) Let V=R². For (11,91), (2, 32) EV and c ER, define addition and scalar multiplication operations by (1, 1) (2, 32) = (1 + 2x2, y1 + 3y2), f(0,0) if c = 0, (cr₁, ) if c# 0. co (₁,1)= Is (V, 0, 0) a vector space over R? If No, please mention each axiom which is not satisfied by these operations.
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